Question

Average Daily Holding cost Solon Container receives 450 drums of plastic pellets every 30 days. The inventory function (drums on hand as a function of days) is \(I(x)=450-x^{2} / 2\) (a) Find the average daily inventory (that is, the average value of \(I(x)\) for the 30 -day period). (b) If the holding cost for one drum is \(\$ 0.02\) per day, find the average daily holding cost (that is, the per-drum holding cost times the average daily inventory).

Short answer

The average daily inventory is 300 drums and the average daily holding cost is $6.00 per day.

Step-by-step solution

Find the average daily inventory

In order to find the average value of a function over an interval, one should integrate the function over the interval and then divide by the length of the interval: \( \frac{1}{b - a}\int_{a}^{b} f(x) dx \)Here, \(f(x) = I(x) = 450 - \frac{x^2}{2}\), \(a = 0\) and \(b = 30\), your integral should be: \( \frac{1}{30}\int_{0}^{30}(450 - \frac{x^2}{2} ) dx \)

Compute the integral

Evaluate the integral \(\frac{1}{30}\int_{0}^{30}(450 - \frac{x^2}{2} ) dx\), Apply the power rule in reverse (which states that the integral of \(x^n\) with respect to \(x\) is \(\frac{x^{n+1}}{n+1}\), along with the linearity of the integral, which allows you to break down the integral of a sum into a sum of integrals. This gives you: \( \frac{1}{30} [ \int_{0}^{30} 450 dx - \int_{0}^{30} \frac{x^2}{2} dx ] \), after computation, we get the average daily inventory equal to 300 drums.

Calculate the average daily holding cost

The average daily holding cost is equal to the average daily inventory multiplied by the holding cost per drum. The holding cost per one drum per day is $0.02. Therefore, the average daily holding cost can be calculated as \( Daily\ Inventory * Holding\ cost\ per\ drum = 300 * $0.02 = $6.00 \) per day. That’s the amount Solon Container needs to pay on average each day to maintain its inventory.

Key concepts

Average Daily Inventory

In an economic context, 'average daily inventory' is a significant figure for businesses to manage resources efficiently. It represents the average number of items in stock on any given day. For example, consider a company that restocks 450 units of a product every 30 days. This company's inventory doesn't remain constant; it fluctuates as products are sold and new shipments arrive.

To compute the average daily inventory, we look at the inventory function, which, in our case, is expressed mathematically as \(I(x) = 450 - \frac{x^2}{2}\), where \(x\) denotes the number of days. To find the average inventory over a 30-day period, a definite integration method is applied to the inventory function, which is integrated over the 30-day interval and then divided by 30, resulting in the average inventory throughout the period. This calculation helps in determining how much stock is held on an average day, which is essential for planning storage space, managing cash flow, and maintaining supply chain operations.

Definite Integration

Definite integration is a cornerstone of calculus with extensive applications in economics, particularly in analyzing inventories and costs. It allows us to calculate the total value or area under a curve within a specific range, which in our context, gives us the total number of items in inventory over a period of time.

Applying this to the inventory function \(I(x) = 450 - \frac{x^2}{2}\), we use definite integration to find the total number of units in inventory from day 0 to day 30. The formula for finding the average of a function over an interval is \( \frac{1}{b - a}\int_{a}^{b} f(x) dx \), where \(a\) and \(b\) define the range of the interval. The integral symbol is indicative of summing all infinitesimally small quantities. After evaluating the integral for our function, we find the total inventory across the days, from which we can then calculate the average daily inventory. It's a vital tool for economic analysis, informing inventory decisions, and budgeting.

Holding Cost Calculation

Understanding 'holding cost calculation' is essential for businesses to comprehend the financial impact of inventory management. Holding costs are the expenses associated with storing inventory over time. These can include storage fees, insurance, depreciation, and risk of obsolescence, among others.

In our problem, the holding cost is given as \(0.02 per drum per day. To find the average daily holding cost, we multiply the average number of drums in inventory by the per-drum daily holding cost. Using the average daily inventory calculated through definite integration, we compute the average holding cost. For Solon Container, with an average of 300 drums in inventory, the calculation is straightforward: \(300\) druoms \(* \)0.02\) per drum, equalling \($6.00\) per day. This informs the company of the daily cost incurred to keep their inventory, enabling better financial planning and cost control strategies. Just like it's crucial to understand the quantity of inventory held on average, it's also necessary to grasp the costs associated with maintaining that inventory.